On uniform log $K$-stability for constant scalar curvature K\"ahler cone metrics

We prove that the existence of constant scalar curvature K\"ahler metrics with cone singularities along a divisor implies log $K$-polystability and $G$-uniform log $K$-stability, where $G$ is the automorphism group which preserves the divisor. We also show that a constant scalar curvature K\"ahler cone metric along an ample divisor of sufficiently large degree always exists. We further show several properties of the path of constant scalar curvature K\"ahler cone metrics and discuss uniform log $K$-stability of normal varieties.